Volume Four of
Essays In Cosmic Archaeology
Essays where anthropological cosmology
meet with cosmology in physics
Genesis Project California 2021!
⋅
⋅
© 2021 By Ian Beardsley"
ALL RIGHTS RESERVED INCLUDING THE RIGHT OF REPRODUCTION IN WHOLE OR IN
PART IN ANY FORM. PUBLISHED BY GENESIS PROJECT
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The Quintessential Elements
Ian Beardsley (Oct 6, 2021)!
Physics, University of Oregon
Genesis Project California 2021
⋅
⋅
of 2 32
1. Abstract We demonstrate the periodic table for artificial intelligence (AI) elements Arsenic
to Iron either elements of semiconductors for transistor technology or conductors for
electrical wire, adhere to a musical scheme. We want to describe life in terms of these AI
elements and show they are both mathematical constructs. The relationship between notes
by music theory, is a mathematical construct. Elements are shown to be in five-fold (fifth
root) relation making them quintessential.!
2. Nickel and Beryllium Alpha decay is the decay of elements where a helium nucleus is
ejected from an element (alpha particles). This happens with atoms heavier than nickel
because the atom has to be heavy enough that the internal repulsion of protons is high enough
that the binding force has trouble holding them together. However, interestingly beryllium-8 is
an exception; it is much lighter than nickel. Beryllium-8 was determined by Fred Hoyle to make
carbon in nuclear synthesis in stars by combining with helium. He sought to find the process
by which carbon is created because the Universe needs to make it if we are to have life as we
know it. !
3. The Theory We pull these AI elements out of the periodic table of the elements to make an
AI periodic table:!
We now notice we can make a 3 by 3 matrix of it, which lends itself to to the curl of a vector
field, by including biological elements carbon C (above Si):!
=!
=!
!
Which resulted in Stokes theorem (Beardsley, Essays In Cosmic Archaeology Volume 3):!
(Ge ⋅ As − Si ⋅ G a)
i + (C ⋅ P)
k
[
(72.64)(74.92) − (28.09)(69.72)
]
i +
[
(12.01)(30.97)
]
k
of 3 32
!
Where!
!
!
!
We were then able to write this with product notation!
!
While we have the AI BioMatrix!
Which we used to formulate a similar equation (Beardsley, Essays In Cosmic Archaeology
Volume 2)!
We can form another 3X3 matrix we will call the electronics matrix (Beardsley, Cosmic
Archaeology, Volume Three):!
!
We can remove the 5th root sign in the above equation by noticing!
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = ex p
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
∇ ×
u = (Ge ⋅ As − Si ⋅ Ga)
i + (C ⋅ P)
k
d
a =
(
zdydz
i + yd yd z
k
)
u = (−C ⋅ P)y
i + (Si ⋅ Ge)z
j + (Ga ⋅ As)y
k
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
of 4 32
!
=(28.085)(72.64)(12.085)(107.8682)(196.9657)=!
!
Where we have substituted carbon (C=12.01) the core biological element for copper (Cu).!
But since we have:!
!
1
We take the ratio and have!
!
Almost exactly 3 which is the ratio of the perimeter of regular hexagon to its diameter used to
estimate pi in ancient times by inscribing it in a circle:!
!
Perimeter=6!
Diameter=2!
6/2=3!
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ Cu ⋅ Ag ⋅ Au
523,818,646.5
g
5
mol
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = 170,535,359.662(g/m ol )
5
523,818,646.5
170,535,359.662
= 3.0716
See Appendix 1
1
of 5 32
!
Thus we have the following equation…!
!
4. Musical Intervals The western tonal system divides up the tonic to double its frequency in
12 tones. But the musical key is defined by the pattern that is only eight of these tones. Thus
at times we must skip a tone (a note). In fact, to comprise the tones of a major or minor key we
play the maximum number of whole steps (the skipping of a note). Since twelve divided by two
is six and we need eight tones two of them must be a half step (where we don’t skip a note). In
the major key this is wholestep—wholestep—halfstep. The minor key begins wholestep-
halfstep.!
We guess that the periodic table of the elements is music (See Figure 1). To do this I am
interested in starting at arsenic (As) because it is a semiconductor doping agent. We then skip
germanium (Ge) an element doped with arsenic to make negative (n-type) germanium, and go
to the next element to the left after the germanium to have gone a whole step and landed on
the other doping agent gallium (Ga) which in contrast makes positive (p-type) germanium. Then
we continue with another whole step to make a major key and it is copper (Cu) the most
abundant practical conductor used for making electrical wire. Now we have to go a half a step
which takes us to nickel (Ni) which like copper is perfect because it is the element beyond
which alpha decay can happen (emits) an alpha particle (helium nucleus) when heated
(Beryllium an exception). Now we go a whole step as is done in the major key and this lands us
on iron (Fe). This is good because stars on the main sequence have luminosities in terms solar
luminosities L related to their mass M in solar masses given by!
!
But we notice the molar mass of gold (Au) divided by iron (Fe) is 3.5 yielding!
"
π = 3.141...
π
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
5
∏
i=1
x
i
L = M
3.5
L = M
Au
Fe
Figure 1: The elements as musical intervals.
of 6 32
We see whether or not we take a whole step in the periodic table for these elements we get a
ratio between successive terms that is always around 107.635. We look at the musical scale
taking C major since the notes are all natural. We have!
C. D. E. F. G. A. B. C.!
We have the halfstep - between E and F - is 106% where for the whole steps it is 112.25%.
The geometric mean between these gives!
!
Which is very close to our 107.635 and we have!
109.08024-107.635=1.44524!
(109.08024)/107.635=1.013427231!
Why look at these elements by molar mass in terms of musical notes? By making the halfsteps
that change in pitch by the same amount for each step as well as these half steps being such
that 12 of them is an octave, then they land on frequencies such that every other note in a
scale makes the interval of a third and three of them defining what is called a triad and they
land on harmonic intervals such as the 4th, and the 5th (See Figure 2). These add more
constructively as vibrating waves than say other intervals that are more dissonant. We can
relate these changes in frequency to changes in string length such as 3/4 and 2/3 which may
have something to offer in terms of physical construction of matter: We see our half step for the
elements occurs for nickel to copper, which is what we want!#
(106)(112.25) = 109.0802457
Figure 2: String length as related to frequency.
of 7 32
5. Elemental Scales So let’s do a more thorough analysis with increase in note frequency
increasing as molar mass frequency increases (See Figures 3 and 4). We have!
!
!
!
We see here the result the ratios between successive elements by molar mass is about the
same as successive half steps between notes (cycles per second). In a minor scale we start at
Fe then skip cobalt (Co) to make a whole step to nickel (Ni). This is a percent change of!
!
We see the whole step from Ni to Fe is 105% about a musical half step like C/B=106%. For
this to be a minor scale the next step is a half step from Ni to Cu, which gives!
!
Our whole steps are E/D=112% and (112+106)/2=109% putting the elemental half step of !
Cu/Ni the average between a musical whole step and a musical half step. In the minor
pentatonic we have whole steps and an interval of a minor third. This is!
!
Thus the minor third for these elements is about a musical whole step (112).!
We see choosing Iron (Fe) is wise as our starting point because as we said for a main sequence
star!
!
And, this makes the last element krypton which being in the last column of the periodic table
(group 18) is an inert gas — so it is like the the note C which is the reference point for music in
the western system of music theory. Krypton being in group 18 and an inert gas because of
this, is an elemental reference point in the periodic table because for example for carbon (C) we
have 18-14=4 valence electrons to acquire noble gas electron configuration. Our pentatonic
scale is the notes!
Kr
Br
=
83.80
79.90
= 104.88
C
B
=
523.25
493.88
= 105.9467887
Br
Si
=
79.90
78.96
= 101.19
B
A
♯
=
49.88
466.164
= 105.9455471
Se
As
=
78.96
74.92
= 105.39
A
♯
A
=
466.164
440
= 105.946
Ni
Fe
=
58.69
55.85
= 105.085
E
D
=
329.628
293.665
= 112.25
Cu
Ni
=
63.55
58.69
= 108.28
F
E
=
349.228
329.628
= 106
Zn
Co
=
F
♯
D
♯
⟶
65.39
58.93
= 111 =
369.944
311.127
= 119
L = M
Au
Fe
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D sharp, F sharp, G sharp, A sharp,…!
Which are elements!
Co, Zn, Ge, Se,…!
And these are the notes outside of the minor scale beginning at Fe. But we can start our minor
pentatonic on Fe, which corresponds to a different key. We have minor pentatonic 2 is:!
D, E, F, G, A,…!
Which are elements!
Fe, Cu, Ga, As,…!
We find that in both pentatonic scales, 1 and 2 the intervals of a minor third (Zn/Co and Cu/Fe)
are very close to the musical whole steps which are are a changes of 112%. Figures 3 and 4…#
of 9 32
!
Figure 3: Molar mass and Hz
of 10 32
!
Figure 4: Minor pentatonics in two different keys.
of 11 32
6. Length and Frequency
Reduce the string length to 2/3 its open length and that is an interval of a fifth. That is a change
in frequency of D=293.665 Hz to B=493.88 Hz. Reduce the string length to 3/4 its open length
and that is a change in frequency from D=293.665 Hz to A=440 Hz, which is an interval of a
fourth. Cut the string length in half and you double the frequency. Which is an octave. We have!
!
!
!
Where is the golden ratio and is the golden ratio conjugate. And,…!
!
Thus, we see string length is inversely proportional to frequency. There are two equations for
string length, which we can examine by looking at a guitar (See figure 5). From the bridge of a
guitar to the fret, where the open string length is s we have:!
!
And!
!
Where s is the string length from the bridge to the nut and the nut is , fret 1 is , fret 2 is ,…!
is the distance from , and is the distance from , and so on…!
The 3/2 is an approximation to the the golden ratio (Phi).!
We can think of electron orbits in a hydrogen atom as frequencies related to length as well
because for a drop in orbit a photon is emitted that has a frequency associated with it. The
change in orbit is like the change in the length of a string. Since the n=3 orbit is a distance of
R3=4.761E-10m from the nucleus and R2=2.116E-10m we have R3/R2=2.25. But the Energy at
E3 is E3=-13.6eV/9, and at E2 it is -13.6eV/4. This is a ratio of 4/9=0.444 and 1/0.444==2.25,
thus there is an inverse relationship between length and frequency in the atom (See Fig. 5).!
1
2
= 2
2
3
=
493.88
391.995
= 1.681 = Φ
Φ =
5 + 1
2
=
1
ϕ
, ϕ =
5 − 1
2
≈ 2/3
Φ
ϕ
3
4
=
440
293.665
= 1.2599 =
4
3
= 1.5
l =
s
2
n/12
l
i
=
s
i
17.817
l
0
l
1
l
2
l
1
l
0
l
2
l
1
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!
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7. Discussion
I find this equation very beautiful!
!
Because on the left side we have what I call the AI matrix:!
!
Because it builds the u vector!
!
And and on the right side we have the what I call the electronics matrix:!
And in that on the left we have a double integral multiplied by pi, over an area and on the right
we have a product operator. Essentially the beauty is in the integral calculus on the left
connecting the AI matrix to the electronics matrix on the right with the product calculus. We
have to ask why this holds for the molar masses of these elements. It has a geometric
representation in that it is flux of the curl of a vector field through a surface with the product of
five elements on the right. This is built from Stokes theorem for these semiconductor and
electronics matrices. What could have brought about the physical properties of these elements
such that this relationship between surface and line holds which we see in its manifestation as!
!
!
π
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
5
∏
i=1
x
i
u = (−C ⋅ P)y
i + (Si ⋅ Ge)z
j + (Ga ⋅ As)y
k
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = ex p
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ C ⋅ Ag ⋅ Au
of 14 32
Where we have the finest transistor elements Si and Ge to the the left of the core biological life
element C and on the right of it the finest conductors for electronics wire Ag, and Au. What
force was behind the creation of these elements such that these theorems hold for them, and
does it serve a purpose, and if it does is there a reason for it?!
This question has been the reason for finding in this paper how the atoms which are elements
might be music theory, because these equations seem to be music.!
8. Conclusion: We may be able to explain the structure of matter at its basis - the elements -
in terms of music theory. The reason perhaps the electron orbits might be described by music
is that musical intervals are in ratios that don’t interfere with one another, which may what is
meant by sonority. Further In our finding of a Stokes theorem representation of semiconductor
and electronics matrices we see the categories of biological and electronics elements may be
mathematical constructs not just chemical. Why?!
They seem to be reconciling!
!
For a hexagon (a is its apothem, P its perimeter) with!
!
For a circle, in that!
!
!
As we show using the most accurate data available in Appendix 1 which are accurate to at
least 3 places after the decimal except germanium."
A =
1
2
aP
A = πr
2
523,818,646.5
170,535,359.662
= 3.0716
3.141 + 3.00
2
= 3.0705
of 15 32
!
Plot of the surface area
of 16 32
9. Atomic Number A very convenient way to estimate pi is with the regular hexagon because
its side is the same as its radius. Thus if its side is one then its radius is one meaning its
permitter is six and its diameter is two giving the integer three even:!
!
We see this can very accurately be approximated by averaging the regular
pentagon with the regular octagon.!
!
The sum of the angles is . .
, , and 54+36=90 where
. a=0.68819096. We have and
then the diameter D is .!
The perimeter P over the diameter is . By similar reasoning we
have for a regular octagon:!
!
. The angle !
Thus,…!
!
Is a regular hexagon.!
We see that the atomic radio of silicon the core element of artificial intelligence (transistor
technology) fits together with the core element of biological life carbon if the silicon is taken as
inscribed in a regular dodecagon and the carbon is taken as inscribed in a regular octagon. We
have:!
, , !
, , !
Apothem: "
A = 180
∘
(n − 2)
A = 180
∘
(3) = 540
∘
540
∘
5
= 108
∘
108
∘
2
= 54
∘
a /s
2
= tan54
∘
2 cos36
∘
= Φ
a
2
+ (s /2)
2
= 0.850650808
D = 1.701301617 ≈ 3
P/D = 2.938926261 ≈ 3
P/D = 3.061457459 ≈ π
22.5
∘
= 0.41421 ≈ 2 − 1
3.061407459 + 2.93892621
2
= 3.00019686 ≈ = 3.00000 = 3
D = 1 + 2x
x
2
+ x
2
= 1
2
2x
2
= 1
2x
2
= 1
x = 2/2
D = 1 + 2
a = (1 + 2)/2 = 1.2071
of 17 32
For a regular dodecagon:!
!
The radius of a silicon atom is Si=0.118nm and that of carbon is
C=0.077nm:!
!
!
!
This has an accuracy %!
It makes sense that we define molar mass in terms of carbon because being element six it can
be thought of in terms of our regular hexagon which defines pi as an integer, the integer 3 (that
is to say as a whole number, 3 is not a fraction in that there is nothing but zeros after the
decimal). Carbon can be thought of in terms of the regular hexagon because it describes the
closest packing of equal radius spheres (a so-called “six-around-one”):!
But though carbon may be six protons, it has six neutrons giving it
a molar mass of 12.01 approximately twice it atomic number of
six. But twelve is our dodecagon which inscribes silicon if it is to
line up with carbon as the regular octagon. And here it is
interesting to note that carbon is made in stars by combining
beryllium-8 with helium, the eight of our regular octagon.!
Silicon is below carbon in group 14 and below that is germanium. Germanium is the other
primary semiconductor element used in transistor technology. In fact it was the first one used
and is called the first generation semiconductor where silicon is called a second generation
semiconductor and is what we mostly use today. Germanium is element 32 and has a molar
mass of 72.64. We have 72.64-32=40.64 giving it 40 neutrons. In considering the weighted
average of germanium most atoms have 42 neutrons (germanium-74). Ge consists of 5
isotopes:!
a =
s/2
tan(θ /2)
=
0.5
ta n(15
∘
)
= 1.866
Si
C
=
0.118
0.077
= 1.532
a
Si
a
C
=
1.866
1.2071
= 1.54585
1.532
1.54585
100 = 99
of 18 32
!
The electronic configuration of germanium is and has a radius of
0.137nm and a Vander Walls radius of 0.211nm. We can predict the atomic numbers of carbon,
silicon, germanium, tin…as we move down group 14 with mathematical patterns combined
with logical statements:!
If n<2!
{!
(n=1,2,3,….)!
!
If n>1; i<3!
{!
(n=2, 3. 4….). (i=1, 2, 3,…)!
!
!
For!
{!
(n=2,3,4,…)!
!
(Ar)(3d )
10
(4s)
2
(4p)
2
Z = 2
n
+ 2
n+1
Z = 2
1
+ 2
2
= 2 + 4 = 6 = carbon
Z = i ⋅ 2
n
+ 2
n+1
Z = 1 ⋅ 2
2
+ 2
3
= 4 + 8 = 12 = silicon
Z = 2 ⋅ 2
3
+ 2
4
= 2 ⋅ 8 + 16 = 16 + 16 = 32 = ger m aniu m
Z = n(3
n
+ 2
n+2
)
Z = 2(3
2
+ 2
4
) = (9 + 16) = 50 = tin
of 19 32
If carbon is octagon (8), silicon dodecagon (12) then germanium is 16-gon. We have…!
!
!
a=2.513669746. !
!
!
a=2.513669646, Ge=0.137nm!
If a1 is octagon a1=1.2071 and a2 is 2.51367 (16-gon) then…!
!
Which is approximately chlorine (Cl=35.45g/mol). If a1 is octagon a1=1.2071 and a2 is
dodecagon a2=1.866 then…!
!
Which is approximately titanium Ti=47.88 then if we average these we have!
!
This is approximately the atomic number of calcium (Ca=40.08 g/mol). Calcium is the primary
component of the mineral component of bone, hydroxyapatite (HA). It does indeed have a
connection to germanium and silicon the primary components of AI circuitry. If we look at the
data for HA, Si, and Ge in terms of density we have!
Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
a =
(s/2)
ta n(θ /2)
=
0.5
ta n(11.25
∘
)
360
16
= 22.5
22.5
2
= 11.25
a
2
= 6.318535592
a
2
+ (s /2)
2
= 2.62915448
D = 5.125830895
P/D = 3.121445152 ≈ π
Ge
x
=
a
2
a
1
x = 72.64
1.2071
2.51367
= 34.88275868g /m ol
x = 72.64
1.2071
1.866
= 46.99021651 ≈ 47g/m ol
34.88275868 + 46.99021651
2
= 40.9364876 ≈ = 41g/m ol
of 20 32
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
G e ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
G e = H A
2 SiG e
Si + Ge
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
G e
=
1
2 + 1
Si ≈ G e − H A
H A ≈
2 SiG e
Si + G e
Si ≈ G e −
2 SiG e
Si + G e
(Si + G e)G e
Si + G e
−
(Si + G e)Si
Si + G e
−
2 SiG e
Si + G e
= 0
G e
2
− 2SiG e − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
of 21 32
And we find this is even more accurate by molar mass:
93% accuracy
We actually find
Si/Ge=
98.78% accuracy
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
G e
≈
1
2 + 1
Si
G e
=
28.085
72.64
= 0.386632709
1
2 + 1
= 0.4142
0.386632709 ≈ 1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
of 22 32
98.75% accuracy
Si/Ge~
An accuracy of 99.95863%
They are almost exactly the same!
10. Silicon and Carbon
We guess that artificial intelligence (AI) has the golden ratio, or its conjugate in its means
geometric, harmonic, and arithmetic by molar mass by taking these means between doping
agents phosphorus (P) and boron (B) divided by semiconductor material silicon (Si) :
Which can be written
We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the
golden ratio conjugate (phi) as well:
So we can now establish the connection between artificial intelligence and biological life:
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
PB
Si
=
(30.97)(10.81)
28.09
= 0.65
2 PB
P + B
1
Si
=
2(30.97)(10.81)
30.97 + 10.81
1
28.09
= 0.57
0.65 + 0.57
2
= 0.61 ≈ ϕ
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
C + N + O + H
P + B + Si
≈ ϕ
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
of 23 32
Which can be written:
Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic
mean:
Which is nice because we can write in the second first generation semiconductor as well
(germanium) and the doping agents gallium (Ga) and arsenic (As):
Where
Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require
doping agents. We now have:
11. Germanium And Carbon
We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and
Phosphorus (P) and we get a similar equation:
,
In grams per mole. Then we compare these molar masses to the molar masses of the
semiconductor material Ge:
PB
[
P
Si
+
B
Si
+ 1
]
+
2 PB
P + B
[
P
Si
+
B
Si
+ 1
]
≈ 2HCNO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ 3HNCO
[
PB +
2 PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
Zn
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
PB
(
Zn
Se
)
+
2 PB
P + B
(
Zn
Se
)
+
P + B
2
(
Zn
Se
)
≈ HNCO
2Ga P
Ga + P
= 42.866
Ga P = 46.46749
2Ga P
Ga + P
1
Ge
=
42.866
72.61
= 0.59
of 24 32
Then, take the arithmetic mean between these:
We then notice this is about the golden ratio conjugate, , which is the inverse of the golden
ratio, . . Thus, we have
1.
2.
This is considering the elements of artificial intelligence (AI) Ga, P, Ge, Si. Since we want to find
the connection of artificial intelligence to biological life, we compare these to the biological
elements most abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O),
phosphorus (P), sulfur (S). We write these CHNOPS (C+H+N+O+P+S) and find:
A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time
using CHNOPS the most abundant biological elements by mass:
Ga P
1
Ge
=
46.46749
72.61
= 0.64
0.59 + 0.64
2
= 0.615
ϕ
Φ
ϕ ≈
1
Φ
Ga P(G a + P) + 2GaP
2(Ga + P)Ge
≈ ϕ
Ga P(G a + P) + 2GaP
2(Ga + P)Si
≈ Φ
CHNOPS
Ga + As + G e
≈
1
2
[
Ga A s +
2Ga A s
Ga + As
+
Ga + As
2
][
Ga
Ge
+
As
Ge
+ 1
]
≈ CHNOPS
[
Ga
Si
+
As
Si
+ 1
]
Ga A s
(
O
S
)
+
2Ga A s
Ga + As
(
O
S
)
+
Ga + As
2
(
O
S
)
≈ CHNOPS
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
Ga A s(G a + As) + 2G a As
2(Ga + As)Ge
≈ 1
of 25 32
We can also make a construct for silicon doped with gallium and phosphorus:
And for germanium doped with gallium and phosphorus:
The Dynamic Construct
Above we see the artificial intelligence (AI) elements pulled out of the periodic table of the elements. As
you see we can make a 3 by 3 matrix of them and an AI periodic table. Silicon and germanium are in
group 14 meaning they have 4 valence electrons and want 4 for more to attain noble gas electron
C + H + N + O + P + S
Ga + As + G e
≈
1
2
(C + N + O + H ) ≈
2(Ga + P)Si
Ga P(G a + P) + 2GaP
(P + B + Si )
HNCO ≈
2(Ga + P)Si
(Ga + P)
[
Ga P +
2GaP
Ga + P
]
(P + B + Si )
HNCO ≈
2(P + B + Si )Si
Ga P +
2GaP
Ga + P
Ga P(G a + P) + 2GaP
2(Ga + P)Ge
≈ ϕ
[
Ga P +
2Ga P
Ga + P
+
Ga + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
Ga
Ge
+
As
Ge
+ 1
]
Ga P
(
B
S
)
+
2Ga P
Ga + P
(
B
S
)
+
Ga + P
2
(
B
S
)
≈ HNCO
of 26 32
configuration. If we dope Si with B from group 13 it gets three of the four electrons and thus has a
deficiency becoming positive type silicon and thus conducts. If we dope the Si with P from group 15 it
has an extra electron and thus conducts as well. If we join the two types of silicon we have a
semiconductor for making diodes and transistors from which we can make logic circuits for AI.
As you can see doping agents As and Ga are on either side of Ge, and doping agent P is to the right of Si
but doping agent B is not directly to the left, aluminum Al is. This becomes important. I call (As-Ga) the
differential across Ge, and (P-Al) the differential across Si and call Al a dummy in the differential because
boron B is actually used to make positive type silicon.
That the AI elements make a three by three matrix they can be organized with the letter E with subscripts
that tell what element it is and it properties, I have done this:
Thus E24 is in the second row and has 4 valence electrons making it silicon (Si), E14 is in the first row
and has 4 valence electrons making it carbon (C). I believe that the AI elements can be organized in a 3 by
3 matrix makes them pivotal to structure in the Universe because we live in three dimensional space so
the mechanics of the realm we experience are described by such a matrix, for example the cross product.
Hence this paper where I show AI and biological life are mathematical constructs and described in terms
of one another.
We see, if we include the two biological elements in the matrix (E14) and and (E15) which are carbon and
nitrogen respectively, there is every reason to proceed with this paper if the idea is to show not only are
the AI elements and biological elements mathematical constructs, they are described in terms of one
another. We see this because the first row is ( B, C, N) and these happen to be the only elements that are
not core AI elements in the matrix, except boron (B) which is out of place, and aluminum (Al) as we will
see if a dummy representative, makes for a mathematical construct, the harmonic mean. Which means we
have proved our case because the first row if we take the cross product between the second and third rows
are, its respective unit vectors for the components, meaning they describe them!
12. The Computation
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
A = (Al, Si, P )
B = (G a, G e, A s)
A ×
B =
B
C
N
Al Si P
G a Ge As
= (Si ⋅ A s − P ⋅ G e)
B + (P ⋅ G a − Al ⋅ A s)
C + (Al ⋅ Ge − Si ⋅ G a)
N
A ×
B = − 145
B + 138
C + 1.3924
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 50g /m ol
of 27 32
And silicon (Si) is at the center of our AI periodic table of the elements. We see the biological elements C
and N being the unit vectors are multiplied by the AI elements, meaning they describe them! But we have
to ask; Why does the first row have boron in it which is not a core biological element, but is a core AI
element? The answer is that boron is the one AI element that is out of place, that is, aluminum is in its
place. But we see this has a dynamic function.
13. The Dynamic Function
The primary elements of artificial intelligence (AI) used to make diodes and transistors, silicon (Si) and
germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an
asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4
valence electrons. Thus to have positive type silicon and germanium, they need doping agents from group
13 (three valence electrons) like boron and gallium, and to have negative type silicon and germanium they
need doping agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the
same period as germanium, boron is in a different period than silicon (period 2) while phosphorus is not
(period 3). Thus aluminum (Al) is in boron’s place. This results in an interesting equation.
The differential across germanium crossed with silicon plus the differential across silicon crossed with
germanium normalized by the product between silicon and germanium is equal to the boron divided by
the average between the germanium and the silicon. The equation has nearly 100% accuracy (note: using
an older value for Ge here, is now 72.64 but that makes the equation have a higher accuracy):
=
0.213714502
2(10.81)/(72.64+28.085)=0.214643832
% accuracy
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 126g /m ol
A ⋅
B = A Bcosθ
cosθ =
6241
6300
= 0.99
θ = 8
∘
A ×
B = A Bsin θ = (50)(126)sin8
∘
= 877.79
877.79 = 29.6g /m ol ≈ Si = 28.09g /m ol
Si(A s − G a) + G e(P − Al )
SiG e
=
2B
Ge + Si
(28.085)(74.9216 − 69.723) + 72.64(30.97376200 − 26.981539)
(28.085)(72.64)
0.213714502
0.214643832
= 99.567
of 28 32
Thus, due to an asymmetry in the periodic table of the elements due to boron we have the
harmonic mean between the semiconductor elements (by molar mass):
This is Stokes Theorem if we approximate the harmonic mean with the arithmetic mean:
We can make this into two integrals:
If in the equation (The accurate harmonic mean form):
We make the approximation
Then the Stokes form of the equation becomes
Thus we see for this approximation there are two integrals as well:
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiGe
Si + G e
∫
S
( ∇ × u ) ⋅ d S =
∮
C
u ⋅ d r
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d x d y ≈
1
Ge − Si
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z ≈
1
3
1
(Ge − Si )
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Ge
B
(P − Al )d x d z ≈
2
3
1
(Ge − Si )
∫
Ge
Si
yd y
Si
B
(As − G a) +
Ge
B
(P − Al ) =
Ge − Si
Ge
Si
dx
x
2SiG e
Si + Ge
≈ Ge − Si
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d yd z =
∫
Ge
Si
d x
of 29 32
14. Conclusion
We have
Where
Gold and Iron are the quintessential elements in that gold is the finest conductor at extreme temperatures,
has had the highest value as a metal for ceremonial crafts and representation of money, since ancient
times and iron has defined the Iron Age where metallurgy and tool making reached its height. Silicon,
germanium, copper, and gold are at the crux electrical and transistor technologies as well as phosphorus,
boron, gallium and arsenic. Their mathematical representations are here connected to the stars and open
up into three dimensional geometry in the form of Stokes theorem as five-fold symmetry. The biological
elements are mathematical constructs as well and describe them. Like the interval of a fifth, musically, it
is .
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al )d ydz =
2
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
(
Si
B
(As − G a) +
Ge
B
(P − Al )
)
d yd z ≈
5
∫
Ge
Si
∫
Ge
Si
∇ ×
v ⋅ d
a
v = (−CP ⋅ y, SiG e ⋅ z, G a A s ⋅ y)
d
a =
(
zd yd z
i + yd yd z
k
)
5
5
∏
i=1
x
i
≈ Fe
5
Si ⋅ G e ⋅ C ⋅ Ag ⋅ Au ≈ Fe
L = M
Au
Fe
Φ
of 30 32
Appendix 1
Ge=72.64!
As=74.9216!
Si=28.085!
Ga=69.723!
C=12.011!
P=30.97376200!
=!
=!
!
!
!
!
!
=154,082,837.980+16,452,521.6822=!
!
=!
(28.085)(72.64)(12.085)(107.8682)(196.9657)=!
!
Where we have substituted carbon C=12.01 for copper Cu. We use Cu, Ag, Au because they
are the middle column of our electronics matrix, they are the finest conductors used for
(Ge ⋅ As − Si ⋅ Ga)
i + (C ⋅ P)
k
[
(72.64)(74.9216) − (28.085)(69.723)
]
i +
[
(12.011)(30.97376200)
]
k
3,484.134569
(
g
mol
)
2
i + 372.025855
(
g
mol
)
2
k
∫
Ge
Si
∫
Ge
Si
(
3,484.134569
(
g
mol
)
2
i + 372.025855
(
g
mol
)
2
k
)
⋅
(
zdydz
i + yd yd z
k
)
∫
Ge
Si
∫
Ge
Si
(
3,484.134569
(
g
mol
)
2
⋅ zdzdy + 372.025855
(
g
mol
)
2
⋅ yd zd y
)
∫
Ge
Si
3,484.134569
(
(72.64 − 28.085)
2
2
)
dy +
∫
Ge
Si
372.025855y ⋅ (72.64 − 28.085)d y
3458261.42924
(
g
mol
)
4
(72.64 − 28.085) + 16575.6119695
(
g
mol
)
3
(
(72.64 − 28.085)
2
2
)
170,535,359.662
(
g
mol
)
5
5
∏
i=1
x
i
= Si ⋅ Ge ⋅ C ⋅ Ag ⋅ Au
523,818,646.5
g
5
mol
5
of 31 32
electrical wire. We use C, Si, Ge because they are the middle column of our AI Biomatrix. Si
and Ge are the primary semiconductor elements used in transistor technology (Artificial
Intelligence) and C is the core element of biological life. We have!
!
!
Perimeter/Diameter of regular hexagon = 3.00!
!
The same value as our 3.0716 if taken at two places after the decimal.!
523,818,646.5
170,535,359.662
= 3.0716
π = 3.141...
3.141 + 3.00
2
= 3.0705
of 32 32
The Author!
of 1 20
Extant Numerology
Ian Beardsley (Oct 20, 2021)!
Physics, University of Oregon
Genesis Project California 2021
⋅
⋅
of 2 20
When doing research certain patterns sometimes show up as a result of the units we use. This
might suggest the units we use have meaning. In a sense they do. I wrote (Beardsley, Essays in
Cosmic Archaeology Volume 2, 2021):!
We only need to explain the significance of the duration of one second. The second does
have meaning especially in connection with life. By dividing the day into 24 hours, the
hour into 60 minutes, and the minute into 60 seconds, the second is 1/86400 of day. By
doing this we have a twelve-hour daytime at spring and fall equinox on the equator, 12
being the most divisible number for its size (smallest abundant number). That is to say that
twelve is evenly divisible by 1,2,3,4,6 which precede it and 1+2+3+4+6=16 is greater than
twelve. As such there is about one moon per 30 days and about 12 moons per year (per
each orbit) giving us a twelve-month calendar. This is all further convenient in that the
moon and earth are in very close to circular orbits and the circle is evenly divisible by 30,
45, 60, and 120 if we divide the circle into 360 degrees which are special angles very
useful to the workings of physics and geometry. Further, the 360 degrees of a circle are
about the 365 days of a year (period of one earth orbit) so as such the earth moves
through about a degree a day in its journey around the sun. Thus, through these
observations down through the ages since ancient times we have constructed the
duration of a second wisely enough to make a lot work together. Now we see 6 protons
and 6 neutrons, which is carbon the core element of biological life on the planet where all
of this came together is deeply connected with the second that defines it all.
I also wrote (Beardsley, Weird Mathematics, 2021)!
We have to ask first, what is a gram? A thousand grams (a kilogram) is the mass of a cube of
1/10 of a meter on each side. And what is a meter? It was originally one ten millionths of the
distance from the North Pole to the equator, and is still close to that. This is the metric system.
It is based on ten. But why? Because we have a base ten counting system; we start with one,
then proceed to 2, then to 3…until we we get to 10 and start over…one zero is ten, one one is
eleven, one two is thirteen, and so on. Many have asked why we have a base ten counting
system and no one knows why, but it is thought to be because we have ten fingers to count on.
Then, we ask, what is a mole? It is a number used in chemistry because atoms, molecules and
compounds occur in large numbers and a dozen just won’t do. Therefore it is a very large
number. But how did we define it? Well carbon is at the core of life and 12 is the most divisible
number for its size in that it is divisible by 1, 2, 3, 4, 6,…evenly. And 1+2+3+4+6=16 is greater
than 12 which makes it the smallest abundant number. For instance four is divisible evenly by
one and two, but one plus two equals 3 which is less than four. Thus, we said carbon has
twelve grams per mole. This allowed us to determine the mole as the number of atoms in 12
grams of carbon, grams defined in terms of water and the earth’s size, The number was
determined by Avagadro and is 6.02E23.
This is best explained by something done by the astronomer Christian Huygens. He noted if pi
squared is 9.87 and we use this as earth acceleration instead of 9.81, then for the period of a
pendulum:
T = 2π
l
g
of 3 20
We have
If l=1 meter then T=2 seconds, one swing to the right is one second. He suggested we define
the meter such that the period is 2 seconds if we use it for l.
I found consecutive integers in gallium and arsenic when taken to two places after the decimal.
(Beardsley, Mathematical Construct in Nature, 2021):
We see the eccentricity is
Which is 9 figures eight of which are the first eight consecutive integers 1,2,3,4,5,6,7,8. The
only number that occurs twice is 2. Kind of interesting.
T = 2
π
2
l
g
= 2 l
1 −
Ga
As
= 1 −
69.72
74.92
= 0.263452781
of 4 20
The Search
We are interested in blocks of elements in the periodic table that can be taken directly out of
the periodic table as 3 by 3 matrices so we can take the cross product with the top row of
elements describing the other two vectors as unit vectors. However, while we can do this with
scandium Sc, Titanium Ti, and vanadium V as aerospace metals describing the two rows below
them the groups 3 to 5, we cannot do this with agricultural elements which seem to be in two
separate 2 by 2 matrices:!
=(22.9897693)(40.08)-(24.305)(39.0983)!
=-28.854 g/mol squared!
The Secondary Nutrients (Mg, Ca, K) and,…!
=(12.011)(30.97376200)-(14.007)(28.085)!
=-21.3607 g/mol squared!
The primary nutrients (N, P)!
g/mol!
g/mol!
(5.372592+4.621768)/2=4.9968~5.00 g/mol=(H+He)=1.01+4.00!
If we consider what I will call the aerospace matrix:!
=!
A=(88.91, 91.22, 92.91)=157.6653!
B=(138.91, 178.49, 180.95)=289.65077!
!
=0.9951!
!
!
(
Na Mg
K Ca
)
(
C N
Si P
)
28.854 = 5.371592
21.3607 = 4.621768
Sc Ti V
Y Zr N b
L a H f Ta
A Bcosθ = A ⋅ B
cosθ = 45,444.4104000/45,667.8755
θ = 5.67
∘
A Bsinθ = 4,512
of 5 20
!
Is between Zn and Ga: (Zn+Ga)/2=(65.39+69.72)/2=67.555 g/mol!
{(91.22)(180.9475)-(92.90637)((178.49)}Sc+{(92.90637)(138.9055)-(88.9084)(180.9475)}Ti!
+{(88.9084)(178.4866)-(91.22)(138.9055)}V=!
(-76.827)Sc+(3,182.547)Ti+(3,197.9983)V!
Which conveniently puts Sc, Ti, and V in the top row as unit vectors. Convenient because they
are aerospace metals as they are light and strong in that scandium, and titanium make light
strong alloys for jet engines and spacecraft, vanadium very rare used as a specialty alloy.!
I already did what I call the electronics matrix and the AiBiomatrix (Beardsley, The
Quintessential Elements 2021) which was the inspiration for this paper because of much
success with it, I had for the electronics matrix:!
=!
A=(106.42, 107.87, 65.39)=165.036!
B=(195.08, 196.97, 200.59)=342.18!
!
!
!
!
Is between Ag and Cd: (Ag+Cd)/2=(107.87+112.41)/2=110.14 g/mol which is down one and to
the left from our (Zn+Ga)/2. That is they are part of a diagonal.!
With Cu, Ag, and Au the malleable, ductile conductors most used for electrical wire. Which I
see now could more aptly be called the metallurgy matrix. The AiBiomatrix was!
!
!
4512 = 67.17g /m ol
Ni Cu Zn
Pd Ag Cd
Pt Au Hg
56,472.0cosθ = 55,124.1476
θ = 12.5432
∘
A Bsinθ = 12,264.348
12,264.348 = 110.74g /m ol
A ×
B =
B
C
N
Al Si P
Ga Ge As
= (Si ⋅ As − P ⋅ Ge)
B + (P ⋅ Ga − Al ⋅ As)
C + ( Al ⋅ Ge − Si ⋅ Ga)
N
A ×
B = − 145
B + 138
C + 1.3924
N
of 6 20
With the middle column carbon the core of biological life, and Si, and Ge the primary
semiconductor elements used in transistor technology. We have!
!
!
!
!
!
!
!
!
!
Is half way between nitrogen and oxygen is (N+O)/2=(14.01+16.00)/2=15.005!
The Result
=15.163 g/mol!
=67.17 g/mol!
=110.74 g/mol!
15.163+67.17+110.74=193.073 g/mol=(Ir+Pt)/2=(192.22+195.08)/2=193.65g/mol!
A ×
B =
B
C
N
Al Si P
Ga Ge As
= (Si ⋅ As − P ⋅ Ge)
B + (P ⋅ Ga − Al ⋅ As)
C + ( Al ⋅ Ge − Si ⋅ Ga)
N
A ×
B = − 145
B + 138
C + 1.3924
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 49.76g/m ol
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 125.5g /m ol
A ⋅
B = A Bcosθ
cosθ =
6240.652
6244.88
= 0.999322965
θ = 2.11
∘
A ×
B = A Bsinθ = (49.76)(125.5)sin8
∘
= 229.924765
229.924765 = 15.163g /m ol
B
C
N
Al Si P
Ga Ge As
Sc Ti V
Y Zr N b
L a H f Ta
Ni Cu Zn
Pd Ag Cd
Pt Au Hg
of 7 20
Is between iridium and platinum. We take the ratio between our determinants, the agricultural
matrices and multiply them by this:!
!
!
(4.621768/5.371592)(193.65)=166.618271306 g/mol!
Where the decimal is 618 271 306!
And , , !
6/3=2 = diameter of a unit regular hexagon. Its perimeter is 6, which gives an approximation to
pi of 3. And, we divide the circle into 360 degrees. !
(0.618)(2.71)(3)=5.02434!
!
The decimal is. 24 15 04!
There are 24 hours in a day and the earth rotates through 360/24=15 degrees per hour. There
are four 7 day weeks in a month!
Conclusion
We can always round our molar mass to two places after the decimal. We can’t always round
better than that. Sometimes we can. The values in the matrices were all rounded to two places
after the decimal. I have been very careful in my rounding throughout all of the calculations, the
results were very interesting: It started with phi ( ) the golden ratio conjugate which is the
inverse of the golden ratio, the first three digits of Euler’s number e, and the approximation of pi
( ) of 3 as given by the regular hexagon which is where the side of a regular hexagon equals its
radius (if the sides are one, its perimeter is 6 and diameter is 2). Multiplying these we get
5.02434 and when we take its square root we get 24 15 04 the hours in a day, the degrees
through which the earth rotates in an hour, and the number of weeks in a month. Our matrices
result in molar masses that don’t exist but are in between those that exist such that they are
not horizontal or vertical in the periodic table, but diagonal (Figure 1)."
21.3607 = 4.621768
28.854 = 5.371592
ϕ = 0.618
e = 2.718
pi ≈ 3
5.02434 = 2.2415004
ϕ
π
of 8 20
"
Figure 1: Diagonal Elements That Don’t Exist (At
least in our Universe)
of 9 20
We can break up our square root into a sum of terms, where each term may by revealing using
Bakshali’s Method for approximating a square root (Figures 2 to 5).!
"
Figure 2: Bakshali’s First Estimate
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"
Figure 3: Bakshali’s Second Estimate
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"
Figure 4: Bakshali’s Third Estimate
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"
Figure 5: Bakshali’s Fourth Estimate!
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Venus
For some time I had been seeing 72 crop up. It is in 2001: A Space Odyssey when HAL, the
ship computer malfunctions and tells David Bowman and Frank Pool that the AE35 unit will fail
in 72 hours (3 days) and as such they will lose contact with Earth. I find this interesting because
HAL is silicon and germanium technology (presumably) and I had found silicon and germanium
very accurately connected to the Venus average orbital distance in AUs (Earth Sun separations)
which is 0.72. Venus has been of great interest to the Russians (they have sent several robotic
ships there) which is because according to Carl Sagan they are interested in it because it is our
sister planet (similar in size and mass) but she underwent a runaway greenhouse effect, which
means she is like a failed Earth. We note here in the Bakshali’s Fourth Estimate it converges on
7200 in the decimal. Four is the fourth planet Mars which I see as representing success in that
it is the terrestrial planet we can colonize. I wrote (Beardsley, Perfect Equations, 2021):!
If I consider my two books, Weird Arithmetic and Weird Calculus, and The Mathematical Nature
of Life, certain equations that I formed in them, while all of them are interesting, they stand out.
In the first book I looked at the connection of artificial intelligence to the planets. Out of all the
equations for the planets, Venus was the most dynamic in form, and the most accurate (Better
than 99%). This equation was:!
!
Where Si is silicon and Ge is germanium, the two principle semiconductor elements used to
make diodes and transistors, and are at the heart of artificial intelligence (AI) circuitry. The 0.72
is the amount of astronomical units in the Venus average orbital distance from the Sun, where
an astronomical unit (AU) is the average earth-sun separation. We can see the validity of this
equation as follows:!
!
Where Si and Ge are in in molar mass (grams/mole).!
Though I did not derive these equations, but guessed at them, it was an educated guess which
proceeded from the argument, that the first planet being the closest to the Sun, sets the idea in
motion, that idea being that its distance is in the simplest expression between Si, and Ge,
possible; the ratio between them. Thus for Mercury ( ) we have!
!
In astronomical units because we take an astronomical unit to be 1 AU at earth orbit, because
the earth is the one planets in the solar system that is highly hospitable to life.!
1
Ge
2
2SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72
1
72.64
2
2(28.0855)(72.64) +
28.0855
3
72.64
1 +
28.0855
2
72.64
2
= 0.722995806
P
1
P
1
=
Si
Ge
of 14 20
To make our guess at the distance of the next planet, we first guess at the simplest idea r(n)=n.
Seeing this does not work we go the next simplest expression r(n)=2n. Seeing this does not
work we go one step higher in complexity and try . We see this does not work either
so we go one step of complexity beyond that and we try . And, we find this works.
But it must be tempered with a factor of 0.3 and and adjusted by 0.4. Thus we have the Titius-
Bode Rule for the distribution of the planets:!
!
!
To get the next equation for . Since we are dealing with a doubling effect we
guess it is , or . Which is close but a little too high. So we guess at something lower
and that it involves twice their product and something big, like the sum of their squares to
reduce the number to prevent the product from being too large. We guess that that value then
is . This is a little too low so we average the two to get exactly the result we need:!
!
Since the next planet, our Earth, must be at a greater value than Venus, and the uppercase
scenario for Venus is , we want to reduce 2SiGe by an amount less than the sum of
their squares, we reduce it by the difference of their squares and get for Earth:!
!
It takes a bit of doing, but the next planet ( ) is:!
!
The next location is the asteroids from 2.2 AU-3.2 AU. Since Mars is the last terrestrial planet
(solid), and the asteroid is a bunch of rocks that could not form into a solid planet and after the
asteroids we have the gas giants Jupiter, Saturn, Uranus, Neptune…and this represents a
flipping around point around the asteroid belt, I started counting again with a new pattern
starting with Jupiter as , and flipped the Earth equation and turned a minus sign to a plus to
obtain the Jupiter equation, which is a quadratic in its simplest form in the numerator, and a
product in its simplest form in the denominator, which is great because it is the first planet after
the asteroid belt and is hence:!
r(n) = n
2
r(n) = 2
n
r(n) = 0.4 + (0.3)2
n
n = − ∞, 0,1, 2,…
P
2
= Venus
2P
1
2
Si
Ge
2SiGe
Si
2
+ Ge
2
P
2
=
1
Ge
2
2SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
2SiGe
Si
2
+ Ge
2
P
3
=
2SiGe
Ge
2
− Si
2
P
4
= Mars
P
4
=
2SiGe
2
(Si − Ge)
2
(Si + Ge)
P
1
of 15 20
!
Now what happens is that a simple pattern forms. We simply multiply Jupiter by 2 to get
Saturn, then by four to get Uranus, and finally by six to get Neptune. We have:!
!
!
!
Thus we have the following table:"
P
1
=
Si
2
+ 2SiGe + Ge
2
SiGe
P
2
= Satur n =
2(Si + Ge
2
SiGe
P
3
= Uranus =
4(Si + Ge
2
SiGe
P
4
= Nept une =
6(Si + Ge
2
SiGe
of 16 20
The lighter an element the more abundant, the heavier, the less abundant. Hydrogen is the
lightest element and the most abundant in the universe. Iron, for example is not that abundant
on earth, but it is heavy. However, something like silicon (the main component of sand SiO2)
and the most abundant in the earth crust, is relatively abundant in the universe. It is no wonder
that it is midway between hydrogen and iron, (1.01+55.85)/2=28.43 g/mol which is very close
to silicon Si=28.09. This may be why I find we can model a hypothetical protoplanetary disc
from the solar system formed in terms of silicon. I wrote (Beardsley, Perfect Equation, 2021):!
The Protoplanetary Disc
But, why describe the orbits of the planets in terms of AI Semiconducting elements? My
answer is to do something cosmic: there is great satisfaction in finding the connection between
two things that seem universes apart. And here I present a reason for looking at such a thing,
by considering the protoplanetary disc from which the planets formed. First we form a table of
the masses of the planets.!
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!
Taking the protoplanetary disc as a thin disc we integrate from its center to the edge, with
density decreasing linearly to zero at the edge. Thus, if the density function is given by!
!
And, our integral is!
!
!
!
The mass of the solar system adding up all the planets yields!
!
That accounts for!
82% of the mass of the solar system not including the sun, that is, of the
protoplanetary disc surrounding the sun.!
Using germanium alone, we get,!
!
If we weight the mixture of silicon and germanium as 1/3 and 2/3, then we have!
!
Which is very close.!
93%!
This is all very good, because I only used the planets and asteroids.!
Si + Ge
2
=
2.33 + 5.323
2
= 3.8265g /cm
3
ρ(r) = ρ
0
(
1 −
r
R
)
M =
∫
2π
0
∫
R
0
ρ
0
(
1 −
r
R
)
rdrdθ
M =
πρ
0
R
2
3
π(3.8265)(7.4 × 10
14
)
2
3
= 2.194 × 10
30
grams
M = 2.668 × 10
30
grams
2.194
2.668
100 =
π(5.323)(7.4 × 10
14
)
2
3
= 3.05 × 10
30
grams
π(4.32467)(7.4 × 10
14
)
2
3
= 2.48 × 10
30
grams
2.48
2.668
100 =
of 18 20
Weighting silicon and germanium as 1/4 and 3/4 we have!
!
Which accounts for!
98%!
Of the mass of the solar system (very accurate).!
This mixture of 1/4 to 3/4 is a combination that exists in the Earth atmosphere which is
approximately the mixture of oxygen to nitrogen. The earth atmosphere can be considered a
mixture of chiefly O2 and N2 in these proportions:!
Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning
the molar mass of air as a mixture is:!
!
By molar mass the ratio of air to H20 (water) is about the golden ratio:!
!
I am not saying the solar system was a thin disk with density of the weighted mean somewhere
between silicon and germanium, but that it can be modeled as such, though if the
protoplanetary disk that eclipses epsilon aurigae every 27 years is any indication of what a
protoplanetary cloud is like, it is a thin disk in the sense that it is about 1 AU thick and 10 AU in
diameter (Kemp, Henson, Kraus, Beardsley, Carrol, Astrophysical Journal, January 1 1986
Letters). This around a star orbiting another star.!
The Babylonian Method
The oldest known method of determining a square root is the Babylonian method. We apply it
to our equation:!
!
The consecutive integers 2, 3, 4, 5, 6. It is actually!
=!
0.2634527818049960944743361359800!
We have!
!
π(4.4 . 57475)(7.4 × 10
14
)
2
3
= 2.623 × 10
30
grams
2.623
2.668
100 =
0.25O
2
+ 0.75N
2
≈ air
air
H
2
O
≈ Φ
1 −
Ga
As
= 1 −
69.72
74.94
= 0.26345
1 − .930592632140950 = 0.0694073678590
x
0
= 0.25 = 1/4
of 19 20
!
!
Which are five consecutive integers in the first five numbers after the second iteration and
Venus is the second planet, The consecutive integers are 2, 3, 4. 5, 6 are Venus, Earth, Mars,
Jupiter, Saturn. The third iteration:!
!
Are the consecutive integers 2, 3, 4, 5, 6, 7, 8 but the 7, 8 are separated from 2, 3, 4, 5. 6 by a
2, Venus, where this third iteration is earth (third planet). The consecutive integers 2, 3, 4, 5, ,6
7, 8 are up to eight there are eight planets in the solar system. It starts at Venus (2) and goes to
Neptune (8).!
If Venus I the second planet and is represented by 3 day which are 3(24hrs/day)=72 and!
Then since the earth is the third planet it might be represented by 4 days are 4(24hrs/day)=96,
then Mars being the fourth planet would be represented by 5(24hrs/day)=120.!
0.72/2=0.36 and arcsin(0.36)=21 degrees is approximately the earth tilt to the ecliptic (225
degrees) and 0.96/2=0.48 and arcsin (0.48)=28.68540201 are consecutive integers!
0, 1, 2, …4, 5, 6,…8!
There are 8 planets but 3 and 7 aren’t in the sequence are Earth and Uranus. We have
1.2/2=0.60 is approximately and arcsin (0.6)=36.87 degrees.!
!
Are the doping agent for!
"
x
1
=
1
2
(
0.25 +
0.069473678590
x
0
)
= 0.263814735718
x
2
=
1
2
(
x
1
+
0.069403678690
x
1
)
= 0.263453029382
x
3
=
1
2
(
x
2
+
0.069403678690
x
2
)
= 0.263452781080
1
Ge
2
2SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72
ϕ = 0.618
1 −
Ga
As
1
Ge
2
2SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
= 0.72
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The Author